Axl - you can click on the arc and drag it. The geometry will change and drag as it is able - if it is not dragging the way you want, then you need to apply additional relations and dimensions to lock down what you don't want to move.

Or you can lock down everything and drive the shape changes by modifying the dimensions.

The black lines mean that those sketch entities are defined and will only move if you change the dimensions. The Blue lines mean that the entities are under defined and you can drag them around,

Apply relations (like perp, horizontal, tangent... etc) by selecting the entities and the pop-up window will have relevant relations that you can apply...

I worked on defining my sketch so everything is locked in place and created a tangent relation. But the curve is still circular and I can't drag anything around. I want them to curve like the red lines I've drawn in Photoshop:

Not really! If you could change the shape of an arc, it would no longer be an arc. It would become a spline. You have to remember that SW is a tool requiring precision for manufacturing processes. Photoshop is a tool for creating images where "close" is good enough.

BTW, splines are often key elements in products created by industrial designers.